What’s the highest population growth rate that the Earth can support?

https://www.reddit.com/r/theydidthemath/comments/3cnmqc/request_if_human_reproduction_continues_at_its/

Q: If human reproduction continues at its current rate, how long until we fill the observable universe?

Starship Troopers by Robert Heinlein has a quote in it about why they fight so hard for every world. It's because, they say, all species are driven to reproduce, and will do so until they fill the space available to them. When all the bug worlds are full, they will take human worlds, even if there have been amicable relations up to that point, simply because bug prosperity is worth more to bugs than human prosperity, and vice versa.

They also drop a quote, that if the human race keeps reproducing at its current rate, it will fill the entire observable universe, packed shoulder to shoulder, in 12,000 years.

This is clearly impossible (there just isn't the mass to fill extragalactic space with humans, or anything else for that matter)

Still, the point kinda stands. How long would it take for his statement to be true? 12,000 years sounds really short for how large space is, but I know exponential growth get bonkers if you let it ramp up enough.

A:

First, consider human dimensions. We shouldn't use pure volume, since the humans will be "shoulder-to-shoulder". Using this reference table, the average human would require volume 1.68x0.6x0.27 m3 , which is about 0.27 m3 .

Thus, we require a total of 1.3x1081 humans to fill up the observable universe (calculation).

Given the current population of 7.2 billion and annual growth rate of 1.12% (source), sustained population growth would fill up the observable universe in 14650 years (calculation).

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As for Heinlein's original answer of 12,000, it's actually pretty close!

Still, it's slightly off. Do you think that's because population growth was significantly higher in the 1950's and he continued that projection?

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Yes, very much so! From Wikipedia and census.gov, the population hit 3 billion in 1959, and the census reported a growth rate of 1.5% in its most recent census (which I'm assuming was all Heinlein had to work with), I get a figure of just above 11000 years. Give the humans slightly more breathing room and one could definitely get a result of 12 millenia.

Some variation (10-20%) arises solely from "shoulder to shoulder" being informally defined.

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See also

http://www.askamathematician.com/2009/11/q-whats-the-highest-population-growth-rate-that-the-earth-can-support/comment-page-1/#comment-502088

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